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Stochastic analysis, fall 2014


Dario Gasbarra


10 sp.


Advanced studies.


Probability Theory  (Todennäköisyysteoria 1, 2).



Weeks 36-42 and 44-50, Tuesday 12-14 in room C124 and Thursday 10-12 in room B120,  with two hours of exercise classes per week.


The course will be given in english or finnish depending on the audience. The materials are written  in english.


The subject of this course is martingale theory and stochastic integration.

0. Introduction: functions with bounded variations, Riemann-Stieltjes integral. Pathwise quadratic variation,  Ito-Föllmer pathwise integral and Ito formula. Brownian motion and its quadratic variation. 

I. Paul Levy's construction of Brownian motion. Gaussian processes. Kolmogorov extension theorem and construction of stochastic process on its canonical space. Kolmogorov continuity theorem.

II. Martingales in discrete time: Conditional expectation, martingale transform, forward and backward martingale convergence theorems, uniformly integrable martingales, square integrable martingales, Doob maximal inequality. Change of measure and  Radon-Nikodym derivative.

III. Continuous  martingales. Ito  isometry. Ito integral and Ito formula. Burkholder Davis Gundy inequality. Local time, Ito-Tanaka formula.

IV Change of measure: Girsanov formula, stochastic exponential, Gronwall lemma. Applications in stochastic filtering.

V. Stochastic differential equations, strong and weak solutions. Applications: Probabilistic solution of partial differential equations. Kakutani's theorem, Feynman-Kac formula.

VI. Ito-Clarck martingale representation. Application:  option pricing in Black & Scholes market model.

Lecture Notes  (last updated  18.11.2014),         Exercises ,    home exam

Old webpages spring 2008 ,  fall 2011 , spring 2013


You pass this course by solving exercises in the weekly tutorial sessions and writing and home exam. 


Karatzas and Shreve Brownian motion and stochastic calculus, Second edition, 1998 Springer.

David Williams: Probability with Martingales (Cambridge Mathematical Textbooks).

Mörters and Peres: Brownian motion, Cambridge 2010.

Daniel Revuz ja Marc Yorin kirjaa "Continuous martingales and Brownian motion", 2nd edition Springer 2005
Dieter Sondermann: Introduction to Stochastic Calculus for Finance: A New Didactic Approach, Springer 2007
Timo Seppääinen:  Basics of Stochastic Analysis, Lecture Notes, University of  Wisconsin-Madison.
Richard Bass:    Stochastic Processes, Cambridge 2011.


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1.Wednesday10-12B322Matteo Marcozzi
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